Average Error: 43.4 → 0.8
Time: 34.5s
Precision: 64
Internal Precision: 1344
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \sin y \cdot \frac{\left(\frac{1}{3} \cdot {x}^{3} + \log \left(e^{\frac{1}{60} \cdot {x}^{5}}\right)\right) + 2 \cdot x}{2} i\right))\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.4

    \[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

  2. Taylor expanded around 0 0.8

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\color{blue}{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right)}}{2} \cdot \sin y i\right))\]
  3. Using strategy rm

  4. Applied add-log-exp0.8

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \color{blue}{\log \left(e^{\frac{1}{60} \cdot {x}^{5}}\right)}\right)}{2} \cdot \sin y i\right))\]

  5. Final simplification0.8

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \sin y \cdot \frac{\left(\frac{1}{3} \cdot {x}^{3} + \log \left(e^{\frac{1}{60} \cdot {x}^{5}}\right)\right) + 2 \cdot x}{2} i\right))\]

Runtime

Time bar (total: 34.5s)Debug logProfile

herbie shell --seed 2018248 
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))